A modal type system for safe distributed computing Giuseppe Primiero FWO - Flemish Research Foundation Centre for Logic and Philosophy of Science, Ghent University Giuseppe.Primiero@Ugent.be http://www.philosophy.ugent.be/giuseppeprimiero/ PCC12, Copenhagen, 17th August 2012
Outline Background 1 Contextual Types with Multi-Modalities 2 The Operational Semantics 3 Conclusions 4 G. Primiero (Ghent University) Modal Type Theory PCC12 2 / 31
Background 1 Contextual Types with Multi-Modalities 2 The Operational Semantics 3 Conclusions 4 G. Primiero (Ghent University) Modal Type Theory PCC12 3 / 31
Logical Approaches to Distributed Programming There is a great variety of systems that approach the issue of distributed programming in view of resource accessibility, safety, security, etc. G. Primiero (Ghent University) Modal Type Theory PCC12 3 / 31
Logical Approaches to Distributed Programming There is a great variety of systems that approach the issue of distributed programming in view of resource accessibility, safety, security, etc. The idea of using Curry-Howard based Semantics to this purpose is natural . . . G. Primiero (Ghent University) Modal Type Theory PCC12 3 / 31
Logical Approaches to Distributed Programming There is a great variety of systems that approach the issue of distributed programming in view of resource accessibility, safety, security, etc. The idea of using Curry-Howard based Semantics to this purpose is natural . . . . . . and it has been very recently combined with the extension of the modal apparatus to gain more control on the notion of resources, their location and accessibility. G. Primiero (Ghent University) Modal Type Theory PCC12 3 / 31
Some relevant works A typed λ -calculus with stationary situations and flowing informations: ◮ [Borghuis and Feijs, 2000]: focus on the representation of the order of commands and recover of data from locations. G. Primiero (Ghent University) Modal Type Theory PCC12 4 / 31
Some relevant works A typed λ -calculus with stationary situations and flowing informations: ◮ [Borghuis and Feijs, 2000]: focus on the representation of the order of commands and recover of data from locations. Type Theories for (safe) distributed and staged computing: ◮ [Davies and Pfenning, 2001], [Jia and Walker, 2004], [Moody, 2003]: represent heterogeneity w.r.t. properties, resources, devices, software, services. G. Primiero (Ghent University) Modal Type Theory PCC12 4 / 31
Some relevant works A typed λ -calculus with stationary situations and flowing informations: ◮ [Borghuis and Feijs, 2000]: focus on the representation of the order of commands and recover of data from locations. Type Theories for (safe) distributed and staged computing: ◮ [Davies and Pfenning, 2001], [Jia and Walker, 2004], [Moody, 2003]: represent heterogeneity w.r.t. properties, resources, devices, software, services. Modal logics for local resources: ◮ [Park, 2006]: distinction between transmission of safe values and safe code. G. Primiero (Ghent University) Modal Type Theory PCC12 4 / 31
Some relevant works A typed λ -calculus with stationary situations and flowing informations: ◮ [Borghuis and Feijs, 2000]: focus on the representation of the order of commands and recover of data from locations. Type Theories for (safe) distributed and staged computing: ◮ [Davies and Pfenning, 2001], [Jia and Walker, 2004], [Moody, 2003]: represent heterogeneity w.r.t. properties, resources, devices, software, services. Modal logics for local resources: ◮ [Park, 2006]: distinction between transmission of safe values and safe code. ILP with ✷ for certified mobile computing: ◮ [Bonelli and Feller, 2009]: code and certificate development; corresponds to a variant of the intensional λ -calculus introduced in [Artemov and Bonelli, 2007]: operational interpretation for remote calls. G. Primiero (Ghent University) Modal Type Theory PCC12 4 / 31
This contribution: meaning of expressions We introduce a polymorphic typed system with multi-modal operators that distinguishes between safe values and safe code (extended from [Primiero, 2012]): G. Primiero (Ghent University) Modal Type Theory PCC12 5 / 31
This contribution: meaning of expressions We introduce a polymorphic typed system with multi-modal operators that distinguishes between safe values and safe code (extended from [Primiero, 2012]): ◮ a i : A says that program a is executed at address i for specification A , producing a safe value; G. Primiero (Ghent University) Modal Type Theory PCC12 5 / 31
This contribution: meaning of expressions We introduce a polymorphic typed system with multi-modal operators that distinguishes between safe values and safe code (extended from [Primiero, 2012]): ◮ a i : A says that program a is executed at address i for specification A , producing a safe value; ◮ x i : A says that code for specification A is validly executable at (and bounded to) address i , producing mobile code; G. Primiero (Ghent University) Modal Type Theory PCC12 5 / 31
This contribution: meaning of expressions We introduce a polymorphic typed system with multi-modal operators that distinguishes between safe values and safe code (extended from [Primiero, 2012]): ◮ a i : A says that program a is executed at address i for specification A , producing a safe value; ◮ x i : A says that code for specification A is validly executable at (and bounded to) address i , producing mobile code; ◮ ✷ i ( A true ) : “ A is valid at every address accessible from i ”; G. Primiero (Ghent University) Modal Type Theory PCC12 5 / 31
This contribution: meaning of expressions We introduce a polymorphic typed system with multi-modal operators that distinguishes between safe values and safe code (extended from [Primiero, 2012]): ◮ a i : A says that program a is executed at address i for specification A , producing a safe value; ◮ x i : A says that code for specification A is validly executable at (and bounded to) address i , producing mobile code; ◮ ✷ i ( A true ) : “ A is valid at every address accessible from i ”; ◮ ✸ i ( A true ) : “ A is executable from address i ”; G. Primiero (Ghent University) Modal Type Theory PCC12 5 / 31
This contribution: meaning of expressions We introduce a polymorphic typed system with multi-modal operators that distinguishes between safe values and safe code (extended from [Primiero, 2012]): ◮ a i : A says that program a is executed at address i for specification A , producing a safe value; ◮ x i : A says that code for specification A is validly executable at (and bounded to) address i , producing mobile code; ◮ ✷ i ( A true ) : “ A is valid at every address accessible from i ”; ◮ ✸ i ( A true ) : “ A is executable from address i ”; ◮ ◦ Γ ⊢ ◦ ( A true ) : contexts describe networks in which code is executed; their internal structure refers to ordered composition of commands. G. Primiero (Ghent University) Modal Type Theory PCC12 5 / 31
Some Features Language in an operational semantics with underlying Curry-Howard isomorphism: ◮ Categorical fragment used to interpret operations with safe values ◮ Functional fragment used to interpret distributed computations with safe code Modification of the standard interpretation of propositional modalities; Modalities are used to reason on the contexts/locations that are safe to the evaluation of the λ -terms. G. Primiero (Ghent University) Modal Type Theory PCC12 6 / 31
This Contribution: sum up The polymorphism and the resulting modal type system are used therefore to explore reasoning about distributed computing, obtaining Code Mobility Rules from corresponding properties of the modal operators. Significant contributions of this work are: the strong assumption on the polymorphic nature of codes and 1 values, understood as locally and globally valid processes; an alternative formulation and interpretation of (multi-)modalities 2 for safe distributed computing; the underlying operational semantics for the interpretation of 3 distributed programs. G. Primiero (Ghent University) Modal Type Theory PCC12 7 / 31
Background 1 Contextual Types with Multi-Modalities 2 The Operational Semantics 3 Conclusions 4 G. Primiero (Ghent University) Modal Type Theory PCC12 8 / 31
Language Definition (Kinds) The set K =: { type , type inf } contains the kind type of all specifications valid by everywhere executable programs, defined by term constructors C ; the kind type inf of specifications valid by locally executable codes, defined by variable constructors V . G. Primiero (Ghent University) Modal Type Theory PCC12 8 / 31
Language Definition (Kinds) The set K =: { type , type inf } contains the kind type of all specifications valid by everywhere executable programs, defined by term constructors C ; the kind type inf of specifications valid by locally executable codes, defined by variable constructors V . Definition (Terms) The set of terms T = {C , V} is given by: constructors C := { a i ; ( a i , b j ); a i ( b j ); λ ( a i ( b j )); < a i , b j > } ; variables V := { x i ; ( x i ( b j )); ( x i ( b j ))( a i ) } . G. Primiero (Ghent University) Modal Type Theory PCC12 8 / 31
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